Pair of Tangents from a Point

Q. Question 10
Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.

Q. Question 6
To draw a pair of tangents to a circle which are inclined to each other at an angle of ${60}^{\circ }$, it is required to draw tangents at endpoints of those two radii of the circle, the angle between them should be
(A) ${135}^{\circ }$
(B) ${90}^{\circ }$
(C) ${60}^{\circ }$
(D) ${120}^{\circ }$

Q.

The pair of tangents are drawn from an external point to a cicle. 2 radii are also drawn from the point of contact of those tangents to the centre. Then the quadrilateral formed by the tangents and the radii can always be inscribed in a circle.

1. True

2. False

Q.

A circle is of the form $x^2+y^2+2gx+2fy+c=0$ and a pair of tangents are drawn from $(x_1,y_1)$ to the circle.The combined equation of tangents is $S{S}_1=T^2.$

Where $S=x^2+y^2+2gx+2fy+c$

$S_1=x_1^2+y_1^2+2g{x}_1+2f{y}_1+c$

$T=x{x}_1+y{y}_1+g(x+x_1)+f(y+y_1)+c$

1. True

2. False

Q. The equations of the tangents drawn from the point (0, 1) to the circle $x^2+y^2-2x+4y=0$ are

1. 2x - y + 1 = 0, x + 2y - 2 = 0
2. 2x - y + 1 = 0, x + 2y - 2 = 0
3. 2x - y - 1 = 0, x + 2y - 2 = 0
4. 2x - y - 1 = 0, x + 2y + 2 = 0

Q. If $L\equiv y=4,$ then the locus of the circumcentre of $△PQR$ is

1. $y-2=0$
2. $x-2=0$
3. $y+2=0$
4. $x+2=0$

Q. The equations of the tangents drawn from the origin to the circle $x^2+y^2-2rx-2hy+h^2=0$ are

1. x = 0, y = 0
2. $(h^2-r^2)x-2rhy=0,x=0$
3. y = 0, x = 4
4. $(h^2-r^2)x+2rhy=0,x=0$

Q. Question 10
Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.

Q. The length of perpendicular from $(4,3)$ to the straight line which makes intercepts $4,3$ on coordinate axes is

1. $\frac{12}{5}$
2. $\frac{5}{7}$
3. $\frac{5}{12}$
4. $\frac{7}{5}$

Q. The three sides of a right-angled triangle are in G .P.. The tangents of the two acute angles may be-

1. $\frac{\sqrt{5}+1}{2}$ and $\frac{\sqrt{5}-1}{2}$
2. $\sqrt{\frac{(\sqrt{5}-1)}{2}}$
3. $\sqrt{5}$ and $\frac{1}{\sqrt{5}}$
4. $\sqrt{\frac{(\sqrt{5}+1)}{2}}$

Q. $C_1$ and $C_2$ are two non concentric fixed circle with radii $R_1$ and $R_2$ such that $R_2$ is contained in $C_1$. A third circle $C$ moves in such a way that it touches $C_1$ internally and $C_2$ externally, then find the locus of its centre.